It follows from Theorem 2.1.12 in [1]} that
\(\begin{alignedat}{-1}&\langle \nabla f_i(\mathbf {x})-\nabla f_i(\mathbf {y}),\mathbf {x}-\mathbf {y}\rangle \\&\ge \frac{\mu L}{\mu +L}\Vert \mathbf {x}-\mathbf {y}\Vert ^2+\frac{1}{\mu +L}\Vert \nabla f_i(\mathbf {x})-\nabla f_i(\mathbf {y})\Vert ^2,\forall i\in \mathcal {V}.\end{alignedat}\) 
